Why a football always has exactly twelve pentagons

Question

A ball's surface is stitched from pentagons and hexagons, with exactly three panels meeting at every corner and every edge shared by exactly two panels. Prove that the number of pentagons is always exactly 12, no matter how many hexagons there are. Then check the classic football (20 hexagons) by counting its corners and edges.

Accepted Answer

Twelve pentagons, always. Set up. Let there be p pentagons and h hexagons. Faces: F = p + h. Each pentagon has 5 edges and each hexagon 6, and every edge is shared by two faces, so the number of edges is E = (5p + 6h)/2. Every corner has exactly three edges meeting there, and every edge has two ends, so the number of corners is V = 2E/3 = (5p + 6h)/3. Euler. For any polyhedron shaped like a sphere, V minus E + F = 2. Substituting: (5p + 6h)/3 minus (5p + 6h)/2 + (p + h) = 2. Multiply by 6: 2(5p + 6h) minus 3(5p + 6h) + 6(p + h) = 12, that is, 10p + 12h minus 15p minus 18h + 6p + 6h = 12, whic…

Why a football always has exactly twelve pentagons

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